arXiv:1206.4398v1 [math.CO] 20 Jun 2012
Distance Powers and Distance Matrices of Integral Cayley Graphs over Abelian Groups Walter Klotz Institut f¨ ur Mathematik Technische Universit¨ at Clausthal, Germany
[email protected]
Torsten Sander Fakult¨at f¨ ur Informatik Ostfalia Hochschule f¨ ur angewandte Wissenschaften, Germany
[email protected]
May 21, 2012 Mathematics Subject Classification: 05C25, 05C50
Abstract It is shown that distance powers of an integral Cayley graph over an abelian group Γ are again integral Cayley graphs over Γ. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.
1
Introduction
Eigenvalues of an undirected graph G are the eigenvalues of an arbitrary adjacency matrix of G. General facts about graph spectra can e.g. be found in [7] or [8]. Harary and Schwenk [10] defined G to be integral if all of its eigenvalues are integers. For a survey of integral graphs see [4]. In [2] the number of integral graphs on n vertices is estimated. Known characterizations of integral graphs are restricted to certain graph classes, see e.g. [1], [13], or [15]. Here we concentrate on integral Cayley graphs over abelian groups and their distance powers. 1
Let Γ be a finite, additive group, S ⊆ Γ, − S = {−s : s ∈ S} = S. The undirected Cayley graph over Γ with shift set (or symbol) S, Cay(Γ, S), has vertex set Γ. Vertices a, b ∈ Γ are adjacent if and only if a − b ∈ S. For general properties of Cayley graphs we refer to Godsil and Royle [9] or Biggs [5]. Note that 0 ∈ S generates a loop at every vertex of Cay(Γ, S). Many definitions of Cayley graphs exclude this case, but its inclusion saves us from sacrificing clarity of presentation later on. In our paper [12] we proved for an abelian group Γ that Cay(Γ, S) is integral if S belongs to the Boolean algebra B(Γ) generated by the subgroups of Γ. Our conjecture that the converse is true for all integral Cayley graphs over abelian groups has recently been proved by Alperin and Peterson [3]. Let G = (V, E) be an undirected graph with vertex set V and edge set E, D a finite set of nonnegative integers. The distance power GD of G is an undirected graph with vertex set V . Vertices x and y are adjacent in GD , if their distance d(x, y) in G belongs to D. We prove that if G is an integral Cayley graph over the abelian group Γ, then every distance power GD is also an integral Cayley graph over Γ. Moreover, we show that in a very general sense distance matrices of integral Cayley graphs over abelian groups have integral spectrum. This extends an analogous result of Ili´c [11] for integral circulant graphs, which are the integral Cayley graphs over cyclic groups. Finally, we show that the class of gcd-graphs, another subclass of integral Cayley graphs over abelian groups (see [13]), is also closed under distance power operations. In our proofs we apply the mentioned result and techniques of Alperin and Peterson. To make this paper more selfcontained and to draw additional attention to this beautiful result, we include a proof reduced to our purposes on a more combinatorial level.
2
The Boolean Algebra B(Γ)
Let Γ be an arbitrary finite, additive group. We collect facts about the Boolean algebra B(Γ) generated by the subgroups of Γ.
2.1
Atoms of B(Γ)
Let us determine the second minimal elements of B(Γ). To this end, we consider elements of Γ to be equivalent, if they generate the same cyclic 2
subgroup. The equivalence classes of this relation partition Γ into nonempty disjoint subsets. We shall call these sets atoms. The atom represented by a ∈ Γ, Atom(a), consists of the generating elements of the cyclic group hai. Atom(a) = {b ∈ Γ : hai = hbi} = {ka : k ∈ Z, 1 ≤ k ≤ ordΓ (a), gcd(k, ordΓ (a)) = 1}. Here, Z stands for the set of all integers. For a positive integer k and a ∈ Γ we denote as usual by ka the k-fold sum of terms a, (−k)a = −(ka), 0a = 0. By ordΓ (a) we mean the order of a in Γ. Each set Atom(a) can be obtained by removing from hai all elements of its proper subgroups. We bear in mind that every set S ∈ B(Γ) can be derived from the cyclic subgroups of Γ by means of repeated union, intersection and complement (with respect to Γ). Thus we easily arrive at the following propositions. Proposition 1. For every a ∈ Γ and every S ∈ B(Γ) holds: Atom(a) ⊆ S or Atom(a) ⊆ S¯ = Γ \ S. Proposition 2. For a ∈ Γ let {U1 , . . . , Uk } be the family of proper subgroups of hai. Then we have Atom(a) = hai \ (U1 ∪ . . . ∪ Uk ). Proposition 3. For an arbitrary finite group Γ the following statements are true: 1. Atom(a) ∈ B(Γ) for every a ∈ Γ. 2. For no a ∈ Γ there exists a nonempty proper subset of Atom(a) that belongs to B(Γ). 3. Every nonempty set S ∈ B(Γ) is the union of some sets Atom(a), a ∈ Γ.
2.2
Sums of Sets in B(Γ)
In this subsection Γ denotes a finite, additive, abelian group. We define the sum of nonempty subsets S, T of Γ: S + T = {s + t : s ∈ S, t ∈ T }. We are going to show that the sum of sets in B(Γ) is again a set in B(Γ). 3
Lemma 1. If Γ is a finite abelian group and a, b ∈ Γ then Atom(a) + Atom(b) ∈ B(Γ). Proof. We know that Γ can be represented (see Cohn [6]) as a direct sum of cyclic groups of prime power order. This can be grouped as Γ = Γ1 ⊕ Γ2 ⊕ · · · ⊕ Γr , where Γi is a direct sum of cyclic groups, the order of which is a power of a prime pi , |Γi | = pαi i , αi ≥ 1 for i = 1, . . . , r and pi 6= pj for i 6= j. Hence we can write each element x ∈ Γ as an r-tuple (xi ) with xi ∈ Γi for i = 1, . . . , r. The order of xi ∈ Γi , ordΓi (xi ), is a divisor of pαi i . Therefore, integer factors in the i-th coordinate of x may be reduced modulo pαi i . The order of x ∈ Γ, ordΓ (x), is the least common multiple of the orders of its coordinates: ordΓ (x) = lcm(ordΓ1 (x1 ), . . . , ordΓr (xr )).
(1)
This implies that all prime divisors of ordΓ (x) belong to {p1 , . . . , pr }. Let a = (ai ), b = (bi ) be elements of Γ. The statement of the lemma becomes trivial for a = 0 or b = 0. So we may assume a 6= 0 and b 6= 0. An arbitrary element w ∈ Atom(a) + Atom(b) has the following form: w = λa + µb, 1 ≤ λ ≤ ordΓ (a), gcd(λ, ordΓ (a)) = 1, 1 ≤ µ ≤ ordΓ (b), gcd(µ, ordΓ (b)) = 1.
(2)
We have to show Atom(w) ⊆ Atom(a) + Atom(b). To this end, we choose the integer ν with 1 ≤ ν ≤ ordΓ (w), gcd(ν, ordΓ (w)) = 1, and show νw ∈ Atom(a) + Atom(b). Case 1.
(p1 p2 · · · pr ) | ordΓ (w).
By gcd(ν, ordΓ (w)) = 1 we know that ν has no prime divisor in {p1 , . . . , pr }. On the other hand all prime divisors of ordΓ (a) and of ordΓ (b) are in {p1 , . . . , pr }. This implies gcd(ν, ordΓ (a)) = 1 and gcd(ν, ordΓ (b)) = 1. Setting λ′ = νλ and µ′ = νµ we achieve gcd(λ′ , ordΓ (a)) = 1, λ′ a ∈ Atom(a), gcd(µ′ , ordΓ (b)) = 1, µ′ b ∈ Atom(b). 4
Now we have by (2): νw = νλa + νµb = λ′ a + µ′ b ∈ Atom(a) + Atom(b). Case 2.
(p1 p2 · · · pr )6 | ordΓ (w).
Trivially, for w = 0 ∈ Atom(a) + Atom(b) we have νw ∈ Atom(a) + Atom(b). Therefore, we may assume w 6= 0. Without loss of generality let (p1 · · · pk ) | ordΓ (w), gcd(ordΓ (w), pk+1 · · · pr ) = 1, 1 ≤ k < r.
(3)
Now (1) and (3) imply w = λa + µb = (λa1 + µb1 , . . . , λak + µbk , 0, . . . , 0), λai + µbi 6= 0 for i = 1, . . . , k.
(4)
By gcd(ν, ordΓ (w)) = 1 we know gcd(ν, p1 · · · pk ) = 1. If even more gcd(ν, p1 · · · pr ) = 1 then we deduce νw ∈ Atom(a) + Atom(b) as in Case 1. So we may assume that ν has at least one prime divisor in {pk+1, . . . , pr }. Without loss of generality let gcd(ν, p1 · · · pl ) = 1, (pl+1 · · · pr ) | ν, k ≤ l < r. We define ν ′ = ν + pα1 1 · · · pαl l .
(5)
If we observe that integer factors in the i-th coordinate of w can be reduced modulo pαi i , then we see by (4): ν ′ w = νw. Moreover, (5) and the properties of ν imply gcd(ν ′ , p1 · · · pr ) = 1. As in Case 1 we now conclude νw = ν ′ w ∈ Atom(a) + Atom(b). Theorem 1. If Γ is a finite abelian group with nonempty subsets S, T ∈ B(Γ) then S + T ∈ B(Γ). Proof. The sets S and T are unions of atoms of B(Γ). S =
k [
Atom(ai ), T =
i=1
l [
Atom(bj ).
j=1
Then we have S+T =
[
(Atom(ai ) + Atom(bj )).
(6)
1≤i≤k,1≤j≤l
According to Lemma 1 the sum Atom(ai ) + Atom(bj ) is an element of B(Γ). Therefore, (6) implies S + T ∈ B(Γ). 5
3
Integral Subsets and Group Characters
Let Γ be a finite, additive group, f : Γ → C a complex valued function on Γ. A subset S ⊆ Γ is called f -integral, cf. our paper [12], if X f (S) = f (s) ∈ Z . s∈S
We agree upon f (∅) = 0. So the empty set is always f -integral. Lemma 2. If all cyclic subgroups of the finite group Γ are f -integral, then every set S ∈ B(Γ) is f -integral. Proof. Suppose that S ∈ B(Γ), S 6= ∅. By Proposition 3, S is the disjoint union of atoms A1 , . . . , Ar of B(Γ). Then we have f (S) = f (A1 )+. . .+f (Ar ). Therefore, it is sufficient to show that every atom is f -integral. According to Proposition 2, every atom A with a ∈ A has a representation A = hai \ (U1 ∪ . . . ∪ Uk ) with certain cyclic subgroups U1 , . . . , Uk of hai. Hence, f (A) = f (hai) − f (U1 ∪ . . . ∪ Uk ). As f (hai) ∈ Z, we may concentrate on f (U1 ∪. . .∪Uk ), which can be evaluated by the principle of inclusion and exclusion (see e.g. [14]). f (U1 ∪ . . . ∪ Uk ) =
k X
(−1)p−1
X
f (Uj1 ∩ . . . ∩ Ujp )
(7)
1≤j1
